Root Clustering for Analytic and their Complexity

Transcription

1 Root Clustering for Analytic and their Complexity Chee Yap Courant Institute, NYU August 5, 2015 Mini-Symposium: Algorithms and Complexity in Polynomial System Solving SIAM AG15, Daejeon, Korea Joint work with Vikram Sharma (IMSc, Chennai) Michael Sagraloff and Ruben Becker (MPI, Saarbruecken)

2 Overview of Talk A. Root Isolation Problem B. Isolating Analytic Roots C. Newton-Bisection Process D. Conclusion

3 Next... Root Isolation Analytic Roots Newton-Bisection Process CONCLUSION [Start] [End]

4 Root Isolation

5 Root Finding Given: F :, and ε > 0 Find: ε-approximations of each root of F

6 Root Finding Given: F :, and ε > 0 Find: ε-approximations of each root of F Remarks: ε-approximation means: ε-disc with exactly one root

7 Root Finding Given: F :, and ε > 0 Find: ε-approximations of each root of F Remarks: ε-approximation means: ε-disc with exactly one root Global/local methods: find all roots/ roots in some B 0

8 Root Finding Given: F :, and ε > 0 Find: ε-approximations of each root of F Remarks: ε-approximation means: ε-disc with exactly one root Global/local methods: find all roots/ roots in some B 0 Special case: real roots

9 Root Finding Given: F :, and ε > 0 Find: ε-approximations of each root of F Remarks: ε-approximation means: ε-disc with exactly one root Global/local methods: find all roots/ roots in some B 0 Special case: real roots Generalized case: analytic or harmonic roots

10 Selective History of Root Finding Computational Version of FTA Classical work Descartes, Newton, Euler, Lagrange, Gauss, Fourier, Sturm, Weierstrass, Vincent, Obreshkorff, Ostrowski, Weyl, Henrici,... Modern work on Complexity Schönhage, Pan, Smale, Bini, Sagraloff-Mehlhorn,...

11 Complexity of Roots Complexity of Root Isolation Benchmark Problem: Isolate all roots of an integer polynomial F

12 Complexity of Roots Best bit-complexity bound for Benchmark Problem: Õ(d 2 (d+ L)) [Schönhage (1982), Pan ( )] where F has degree d and L-bit coefficients. This bound is called near optimal (Pan)

13 Complexity of Roots Until now: all near optimal bounds are based of variants of Circle Splitting Method [Schönhage] Recent New Algorithms based on Subdivision: can now reach near optimal bounds are local methods seems quite practical [Kobel (2012), Kamath (2010)]

14 Complexity of Roots Until now: all near optimal bounds are based of variants of Circle Splitting Method [Schönhage] Recent New Algorithms based on Subdivision: can now reach near optimal bounds are local methods seems quite practical [Kobel (2012), Kamath (2010)] Real Roots: [Sagraloff (2014), Mehlhorn-Sagraloff (2015)]

15 Complexity of Roots Until now: all near optimal bounds are based of variants of Circle Splitting Method [Schönhage] Recent New Algorithms based on Subdivision: can now reach near optimal bounds are local methods seems quite practical [Kobel (2012), Kamath (2010)] Real Roots: [Sagraloff (2014), Mehlhorn-Sagraloff (2015)] Complex Roots: this talk

16 Overview of Subdivision Methods Sturm Method Non-adaptive, impractical for large coefficients Descartes Method [Akritas-Collins(1976), Rouillier-Zimmerman (2003)] Near-optimality [Mehlhorn-Sagraloff (2015)] Evaluation Method Real Roots: [Burr-Krahmer-Yap (2009), Sharma-Yap(ISSAC 12)] Complex Roots: [Sagraloff-Yap (ISSAC 11)] Analytic Roots: [Sagraloff-Sharma-Yap (CiE 13)] this talk

17 Goals GOALS of this TALK What techniques are needed beyond algebraic roots? How can Newton iteration be introduced to achieve near optimal bounds?

18 Next... Root Isolation Analytic Roots Newton-Bisection Process CONCLUSION [Start] [End]

19 Analytic Roots

20 Tools Tools Beyond algebraic roots How to represent analytic functions? How to handle multiple roots? Replacement for C 1 (monotonicity) predicate? (Pellet) Replacement for sign evaluation? (Soft sign)

21 Specifying Analytic Functions How is an Analytic Function Given? Interval Functions Given F :, its interval analogue is F :.

22 Specifying Analytic Functions Call F a box function for F if: (1) Conservative: F(B) F(B) (2) Convergent: If lim i B i p as i, then lim i F(B i ) F(p)

23 Specifying Analytic Functions An analytic function F is given by having box functions for all its derivatives, F (i) (z) (i 0)

24 Root Clusters Root Clusters Descartes, EVAL and CEVAL are conditional algorithms conditioned on having only simple zeros! To be unconditional, we introduce clusters A cluster of k roots does not distinguish one root of multiplicity k from k simple roots

25 3D Root Clusters D no roots µ roots A disk D is isolating for F if 3D\ D has no roots of F The roots in D, if non-empty, forms a (root) cluster.

26 Root Clusters LEMMA (Naturalness): Any 2 root clusters are disjoint or has a containment relationship. There are at most 2n 1 clusters contained any set of n roots.

27 Root Clusters So root isolation becomes: Root clustering problem for F: Given a box B 0. Find an ε-isolating system of clusters for B 0.

28 Pellet Predicates For k 0 and reals r,k 1, define C k (m,r,k): F k (m) r k > K F i (m) r i i k where F i (m):= F(i) (m) i! are Taylor coefficients Write C k (D,K if D = Disc(m,r). Write C k (D if K = 1.

29 Pellet Predicates LEMMA [Pellet (1881)] If C k (m,r) holds then the disk D m (r) contains exactly k roots of F. Proof by application of Rouché s Theorem. Pellet s predicate is non-effective if F is analytic

30 Darboux s Theorem To make Pellet effective: THEOREM [Darboux (1876)] Let F : D 0 be analytic in an open disk D 0. Let a,b D 0 and k 0. Then there exists 0 Θ 1and ω, ω 1such that, for h:= b a and ξ := a+θ(b a), F(b)= k F ν (a)h ν + ωh k+1 f k+1 (ξ). ν=0 [Prashant Batra (MCS 2009)]

31 To make Pellet effective: Box Version of C k (m,r,k) C k (m,r,k): F k (m) r k > K Darboux s Theorem ( ) k 1 F i (m) r i + F k+1 (D m (r)) r k+1 i=0 where F k+1 (D) is some box function for F k+1 (z). We have the analogue of Pellet s Theorem for this Interval Version.

32 Exact A-EVAL We design an Exact A-Eval for analytic root clustering. Define: firstc(b,n) to return the smallest k = 0,...,N such that D(2k B) is isolating and contains k roots; else, return 1. To verify that D(2k B) is isolating, we can check the predicates C k (2k B) and C k (6k B).

33 Exact A-EVAL Q 0 {B 0 },Q 1 /0, Q out /0 0. WHILE (Q 0 is non-empty) B Q 0.pop() k firstc(b,n) 1. IF k < 0, split B and push its 4 children into Q ELIF 1 k N, Q 1.push(B,k) 3. WHILE (Q 1 is non-empty) (B,k) Q 1.pop() 4. If (B,k) does not conflict with any pair in Q out, 5. Q out.push(b,k) RETURN Q out

34 Soft A-EVAL The problem with Exact A-EVAL is that it uses exact evaluation of the predicate C k (B). We consider the soft version denoted Replace C k (B,K) in Exact A-EVAL by Ck (B). Ck (B,K).

35 Soft A-EVAL General idea of soft comparison A: B Returns where A B means that A>B or A<B or A B. 1 A<B < 2A. 2 The result is undefined if A = B = 0 and is indeterminate when A B and A B.

36 Soft A-EVAL In the soft version of C k (m,r), the comparison > in C k (m,r,k): F k (m) r k > K ( ) k 1 F i (m) r i + F k+1 (D m (r)) r k+1 i=0 uses a p-bit approximation of the left and right hand side where p= lgr If the test fails, future subdivision ensures that the precision will increase.

37 Soft A-EVAL LEMMA: (i) If Ck (B) succeeds, then C k (B) succeeds. (ii) If C k (B,2) succeeds, then Ck (B) succeeds. Because of this lemma, we use K = 2 in Soft A-EVAL to ensure termination.

38 Next... Root Isolation Analytic Roots Newton-Bisection Process CONCLUSION [Start] [End]

39 Newton-Bisection Process

40 Newton-Bisection Combining Newton with Bisection Newton converges quadraticaly, but conditional success Bisection converges linearly, but has unconditional success

41 Newton-Bisection Combining them is an old idea: E.g., Dekker (1950s), Brent (1960s) in root isolation

42 Newton-Bisection Recently, Abbot introduced a more sophisticated form called Quadratic Interval Refinement (QIR) It was combined with Descartes Method to achieve the near optimal complexity for real root isolation [Sagraloff (2013), Mehlhorn-Sagraloff (2015)]

43 Newton-Bisection We now combine it with Pellet s Test to achieve the near optimal complexity for complex root isolation

44 NB Process The Newton-Bisection Process State (e,n) (error and speed) Transition relation (e,n) (e,n ) if e e/2 2n, e/2 2n 1 n = n+1 e e/2, e/2 2n < 1 n = n 1 (Newton Step) (Bisection Step)

45 NB Process An NB Process is a sequence s=(s 0,s 1,...,s k ) such that s i s i+1. A state (e, n) is small if e 1. THEOREM: If the NB Process has no small states, then its length it at most n 0 + 2lglg(e 0 )+2 where the initial state is(e 0,n 0 ).

46 Application Application Maintain connected components of subdivision boxes; all boxes in a component R having same width w R. The component is contained in an enclosing disc D R with center x R.

47 Application For component R, maintain the state (w R,n R ): initially n R = 4. Apply Pellet s test C k (D R ) and C k (8D R ) holds, for each k = 0,1,...,n.

48 Application This is a supercharged test, using Graeffe s acceleration on the polynomial F

49 Application If a k is detected, we apply the order-k Newton step: x R x R k F(x R) F (x R )

50 Application Check if Pellet test C k (x R,n2 R ) succeeds. (Newton Step) If so, we replace n R by nr 2 ; and update the connected components to those boxes intersecting (x R,n2 R ). (Bisection Step) If not, we replace it by n R, and bisect each box B in C, discarding children that satisfy C 0 (B), and recompute the connected components.

51 Next... Root Isolation Analytic Roots Newton-Bisection Process CONCLUSION [Start] [End]

52 CONCLUSION

53 Open Problems and Directions Complexity analysis analytic roots Extensions: harmonic roots Implementation: how practical is Newton-Pellet isolation?

54 Thanks for Listening! A rapacious monster lurks within every computer, and it dines exclusively on accurate digits. B.D. MCCULLOUGH (2000)

55 Extra Slides